to say Trajan conquered Parthia is giving him more credit than he deserves, imho.
From Jgnat's Roman History link:
But by AD 113 affairs in the east again awakened his military ambitions.
The Euphrates had long been the vaguely acknowledged boundary between the Roman and Parthian dominions, but both empires claimed he northern kingdom of Armenia as a client state.
When the Parthian king Chosroes set up a king of his own on the throne of Armenia it was excuse enough for Trajan to begin a project of yet more military expansion.
In AD 113 he set his armies in motion and proceeded to the east to take command in person.
Chosroes tried to sue for piece, offering to set a new king in Armenia, a certain Parthamasiris, instead of the one Trajan initially took objection to, but it was not enough for the Roman emperor.
Trajan advanced meeting no resistance, till he reached the borders of Armenia. Parthamasiris came in person to plead for an end to hostilities, but only to be told that Armenia was no longer a kingdom but a Roman province, and that he should leave. The circumstances in which Parthamasiris was killed almost immediately after are obscure, but they certainly could not speak well for Trajan.
Armenia with Mesopotamia was secured, but Parthia was the emperor's real objective. Operations however were delayed till AD 116 owing to the need for creating some organization, and then due to the havoc wrought by a terrific earthquake at Antioch, in which Trajan himself barely escaped with his life. Then came a great campaign over the Tigris, the passage of which in the face of an active foe was no easy task, and the advance to Susa, the last triumphant achievement.For in the rear of the victorious armies revolt broke out in the annexed territories. Trajan was obliged to retreat with the enemy behind him, not in front of him, and his own health had at last broken down. He was indeed only checked, by no means defeated, but he saw at least that his dream of recreating the achievements of Alexander the Great could never be accomplished.His health deteriorating rapidly he started on his way back home to Rome, but died on the way in Cilicia (AD 117), having left his chief-of-staff, Publius Aelius Hadrianus, in charge of the inconclusive situation in the east.
----So, yes, Trajan cross the Tigris and won battles, however, he didn't technically subdue the Parthian kingdom. Rome never actually held or administered the land.